Question: Express $1.\overline{03}$ as a reduced fraction, given that $0.\overline{01}$ is $\frac{1}{99}$.
Solution: We know that $1.\overline{03} = 1 + 0.\overline{03}$.  Since $0.\overline{03}$ is three times greater than $0.\overline{01}$, we have $0.\overline{03} = 3 \cdot \frac{1}{99} = \frac{3}{99}$.  This simplifies to $\frac{1}{33}$.  Then, adding one to this fraction, we get $1 + \frac{1}{33} =$ $\boxed{\frac{34}{33}}$.